I'm Trey. My friend Serena and I are working at an archeological dig. Sometimes the crew finds some pretty cool stuff.

Yesterday, we found this stone called a voussoir... vo-us-soir...vo-usso-ir..vou-ssoir...

It's a stone shaped like a trapezoid.

What, you might ask, is a voussoir? The Romans created arches by stacking these trapezoidal stones together into a semi-circle.

The problem is, we want to know how many stones were in the original arch, but, we only found this one stone.

Here's what we know. The stone has a trapezoidal face with legs that are the same size, and each of the acute angles measure 84 degrees.

Serena guessed that there were 9 stones in the original arch. She drew this great picture to prove it.

But I'm not convinced Serena drew the trapezoid with the correct angles. I think we need to recalculate the arch, based on the angles of our stone.

Is Serena right? Were there 9 stones in the original arch?

ABOVE IS THE TRANSCRIPT OF THE VIDEO. IT HAS NOTHING TO DO WITH THE ASSIGNMENT.

The Students' Conjectures:During an archeological dig, the students found avoussoirfrom a semicircular stone arch. Avoussoiris a trapezoidal (wedge-shaped) stone used to build an arch or vault.

1. What are the students trying to find out? (1 point)

2. Complete the table to summarize each student's conjecture: (2 points: 1 point for each row)

Student	Conjecture
Trey
Serena

Analyzing the Data:

3. Draw a picture that shows the trapezoidal stone and the information that you know: (1 point)

4. Is this is an isosceles trapezoid? How do you know? (1 point)

Understand the problem: Measuring an existing arch.

Part 1: Serena's drawing

Follow these steps to find the angles of the trapezoids in Serena's drawing. If the angles of her trapezoid match the angles of the trapezoid stone, then her drawing is correct.

5. Find the measure of angleBAC. Hint: What is the sum of the measures of the vertex angles of the nine congruent isosceles triangles? (1 point)

6. TriangleABCis an isosceles triangle. Find the measure of each base angle. (2 points)

7. Use the base angles of triangleABCto find the measure of angleCBD. (1 point)

8. Find all the angle measures of trapezoidBCED: (2 points)

Solving the problem: How many stones were in the original arch?

9. Compare these angle measures to the angles of the trapezoid stone. What can you conclude about Serena's theory that the arch has nine stones? (2 points)

Part 2: Work backwards to find the number of stones that were in the original arch.

10. TrapezoidEFGHrepresents the trapezoid stone that Trey found. Use your picture from Question 3 to find the measures of the interior angles of trapezoidEFGH.(1 point)

11. Imagine that the stone is part of an arch, and that there is an isosceles triangleJEFformed by extending sidesGEandHFdownward. Use the measure of the angles of the trapezoid to find to measure of the base anglesJEFandJFE. Add these measurements to your drawing.(2 points, one for each angle measure)

12. Find the measure of the vertex angleEJF.(1 point)

13. The isosceles triangleEJFis one of an unknown number of isosceles triangles from the original arch. Each isosceles triangle had the same vertex angle as triangleEJF. The sum of the vertex angles equals 180. (2 points, one for the correct equation, one for the correct solution.) Letx= the number of isosceles triangles. Fill in the equation with your answer from Q12 and solve forx. ___ x= 180

14. There is one stone for each isosceles triangle. How many stones were in the original arch? (1 point)